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Tame arrangements

Takuro Abe

TL;DR

The paper develops a foundational framework for tame hyperplane arrangements, introducing an addition-deletion theory that enables constructing tame configurations and proving that tameness is preserved under controlled local freeness. It extends tameness to multiarrangements, establishing Ziegler restriction tameness and a Yoshinaga-type criterion that links the tameness of an arrangement to that of its Ziegler restriction, with distinctions for low and high ambient dimensions. Generic and inductively defined tameness classes are shown to be combinatorially determined, providing constructive, lattice-based methods to generate tame arrangements. Together, these results connect tameness with local freeness, restriction theory, and combinatorial data, expanding the toolkit for studying Milnor fibers, master functions, and related algebraic-geometric structures.

Abstract

Tame arrangements were informally introduced by Orlik and Terao for the study of Milnor fibers of hyperplane arrangements. After that, tame arrangements have been applied to a lot of researches on arrangements including freeness, master functions and critical varieties, Solomon-Terao algebras, D-modules, Bernstein-Sato polynomials and likelihood geometry. Though arrangements are generically tame, the research on tame arrangements themselves have been only few. In this article we establish foundations for the research of tame arrangements. Namely, we prove the addition theorem for tame arrangements, Ziegler-Yoshinaga type results for tameness and combinatorially determined tameness.

Tame arrangements

TL;DR

The paper develops a foundational framework for tame hyperplane arrangements, introducing an addition-deletion theory that enables constructing tame configurations and proving that tameness is preserved under controlled local freeness. It extends tameness to multiarrangements, establishing Ziegler restriction tameness and a Yoshinaga-type criterion that links the tameness of an arrangement to that of its Ziegler restriction, with distinctions for low and high ambient dimensions. Generic and inductively defined tameness classes are shown to be combinatorially determined, providing constructive, lattice-based methods to generate tame arrangements. Together, these results connect tameness with local freeness, restriction theory, and combinatorial data, expanding the toolkit for studying Milnor fibers, master functions, and related algebraic-geometric structures.

Abstract

Tame arrangements were informally introduced by Orlik and Terao for the study of Milnor fibers of hyperplane arrangements. After that, tame arrangements have been applied to a lot of researches on arrangements including freeness, master functions and critical varieties, Solomon-Terao algebras, D-modules, Bernstein-Sato polynomials and likelihood geometry. Though arrangements are generically tame, the research on tame arrangements themselves have been only few. In this article we establish foundations for the research of tame arrangements. Namely, we prove the addition theorem for tame arrangements, Ziegler-Yoshinaga type results for tameness and combinatorially determined tameness.

Paper Structure

This paper contains 5 sections, 38 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Addition-deletion theorem for tameness
  4. Multiarrangements and tameness
  5. Inductive tameness

Key Result

Theorem 1.4

Assume that ${\mathcal{A}}$ is locally free and $\mathop{\mathrm{pd}}\nolimits_S \Omega^1({\mathcal{A}})=1$. Then ${\mathcal{A}}$ is tame. Explicitly, it holds that $\wedge^p \Omega^1({\mathcal{A}})=\Omega^p({\mathcal{A}}),\ \mathop{\mathrm{pd}}\nolimits_S \Omega^p({\mathcal{A}})=p$ for $0 \le p \le

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Theorem 5.3, MS
  • Theorem 1.5: Addition theorem for tameness I
  • Theorem 1.6: Addition theorem for tameness II
  • Example 1.7
  • Theorem 1.8: Restriction theorem for tameness
  • Theorem 1.9
  • Theorem 1.10
  • ...and 48 more